Algebra and Topology at the University of Udine

*Sala Pasolini, Palazzo Garzolini di Toppo-Wasserman, Udine*

Conference **Algebra, Topology, and Dynamical Systems**

A conference in honor of Dikran Dikranjan on the occasion of his 70th birthday

*Simone Virili*
**Intrinsic entropy and multiplicity of Z[X]-modules**

Abstract: It is a well-established way of thinking about modules over the polynomial ring Z[X] as pairs (G,f), where G is an Abelian group and f is a distinguished endomorphism of G.
In this way, the intrinsic entropy (introduced in [1]) - an invariant of endomorphisms of Abelian groups - can be used to define an invariant of the category of Z[X]-modules.
On the other hand, in Chapter 7 of his famous book "Lessons on rings, modules and multiplicities" [2], D.G. Northcott introduced,
for a commutative ring R and an element r in R, the concept of r-multiplicity of an R-module M.
As P. Vámos noted a few years ago, if one considers the X-multiplicity of Z[X]-modules, this gives an invariant of the category of Z[X]-modules with very nice properties,
which are formally similar to those of the intrinsic entropy. In this talk we will explore the connection between these two invariants.

Bibliography:

[1] D. Dikranjan, A. Giordano Bruno, L. Salce, and S. Virili, *Intrinsic algebraic entropy*, Journal of Pure and Applied Algebra 219, no. 7 (2015) 2933-2961.

[2] D. G. Northcott, *Lessons on rings, modules and multiplicities*, Cambridge University Press (1968).

*University of Lodz, Poland*

Conference **Dynamics of (Semi-)Group Actions**

*Ilaria Castellano (University of Milano-Bicocca, Italy)*
**The inert subgroups of the Lamplighter Group**

*Hans-Peter Künzi (University of Cape Town, South Africa)*
**Splitting ultra-metrics via T0-ultra-quasi-metrics**

Abstract: We report on joint work with Conradie, Gaba and Yildiz. Given a T0-ultra-quasi-metric u on a set X, we write us for its symmetrization.
Many of our results are based on the observation that given a T0-ultra-quasi-metric u on a set X there exists a T0-ultra-quasi-metric v on X such that v≤u, vs=us and the specialization order of v is linear.

*Francesco Russo (University of Cape Town, South Africa)*
**Subgroup commutativity degree of profinite groups**

Abstract: In the present talk it will be introduced a probability measure, which counts the pairs of closed commuting subgroups in infinite groups. This measure turns out to be an extension of what was known in the finite case as subgroup commutativity degree.
After a short survey of the known results in the finite case, the profinite case will be discussed with details.
The so called topologically quasihamiltonian groups are described by the case of subgroup commutativity degree equal to one.
This is a joint work with Eniola Kazeem.

*Lydia Aussenhofer (University of Passau, Germany)*
**Mackey's problem - its historical background and a final solution**

Abstract: For a locally convex vector space (V,τ) there exists a finest locally convex vector space topology μ such that the topological dual spaces (V,τ)' and (V,μ)' coincide algebraically.
This topology is called Mackey topology.
If (V,τ) is a metrizable locally convex vector space, then τ is the Mackey topology.
In 1995 Chasco, Martín Peinador and Tarieladze asked the following question: Given a locally quasi-convex group (G,τ), does there exist a finest locally quasi-convex group topology
μ on G such that the character groups (G,τ)^ and (G,μ)^ coincide? In this talk we give examples of topological groups which have a Mackey topology, and we present a group which has no Mackey topology.

*Sala Pasolini, Palazzo Garzolini di Toppo-Wassermann, Udine*

Conference **Una Giornata per Silvia**

*Ilaria Castellano (University of Southampton, UK)*
**Topological entropy of linearly compact vector spaces and left Bernoulli shifts**

*Pratulanda Das (Jadavpur University, Kolkata, India)*
**Ideal convergence and some observations**

Abstract: We will briefly discuss the motivation behind the notion of ideal convergence and some results associated with the notion. In particular we will also talk about the role of P-ideals and like in the results.

*Aula 8, Palazzo Antonini, Udine*

Conference **Dynamical methods in Algebra, Geometry and Topology**

*Pawel Grzegrzolka (University of Tennessee, Knoxville, TN, USA)*
**Coarse proximity and proximity at infinity**

Abstract: Coarse topology (i.e., large scale geometry) is a branch of mathematics investigating large-scale properties of spaces. While classical topology is primarily concerned with what happens on the small-scale
(e.g., limits, continuity), coarse topology focus predominantly on what happens "on the large-scale" (e.g., asymptotic dimension, coarse equivalence of spaces).

The idea of translating a small-scale world to its large-scale counterpart has been extensively explored by coarse topologists. In this talk, we will focus on
coarsening the notion of proximity. We will start with reviewing the notion of a proximity space and introducing the definition of a metric coarse proximity.
After investigating a few properties of this relation, we will generalize the metric case to obtain coarse proximities on any set with bornology.
Then we will proceed to show the existence of the category of coarse proximity spaces whose morphisms are closeness classes of coarse proximity maps.
We will conclude with the construction of a proximity space at infinity - a coarse invariant of unbounded metric spaces and a link between the small-scale and the large-scale worlds.

No prior knowledge of coarse topology will be assumed. Familiarity with metric spaces is desirable. This is joint work with Jeremy Siegert.

*Andrzej Bis (University of Lodz, Poland)*
**Some estimations of topological entropy of a group**

*Antongiulio Fornasiero (University of Firenze, Italy)*
**Algebraic entropy**